On non-linear discrete boundary-value problems and semi-groups

The following discrete boundary - value problem for non-linear system $x_k=\varphi_k(x_{k-1},y_k),\,y_{k-1}=\psi_k(x_{k-1},y_k)$, $k=\overline{1,\,N},\,\,N<\infty,\,\,x_0=a,\,\,y_N=b, $ is considered. Here the functions $\varphi_k(x,y),\,\psi_k(x,y)\geq0$ are monotone with respect to arguments $ x,\,y\geq0$, satisfing the condition of dissipativity or conservativity:

$ \varphi_k(x,y)+\psi_k(x,y)\leq x+y+\gamma_k$, as well as two simple additional conditions. A relation of this problem with multistep processes is demonstrated. Existence and uniqueness of minimal solution of problem is proved. A semi-group approach to solving of problem is developed. The approach is adjoined with V.~Ambartsumian Principle of Invariance and R.Bellman method of

On the Central Limit Theorem for Toeplitz Quadratic Forms of Stationary Sequences

Let $X(t),$ $t = 0,\pm1,\ldots,$ be a real-valued stationary Gaussian sequence with spectral density function $f(\Lambda)$. The paper considers a question of applicability of central limit theorem (CLT) for Toeplitz type quadratic form $Q_n$ in variables $X(t)$, generated by an integrable even function $g(\Lambda)$. Assuming that $f(\Lambda)$ and $g(\Lambda)$ are regularly varying at $\Lambda=0$ of orders $\alpha$ and $\beta$ respectively, we prove CLT for standard normalized quadratic form $Q_n$ in the critical case $\alpha+\beta=1/2$.

We also show that CLT is not valid under the single condition that the asymptotic variance of $Q_n$ is separated from zero and infinity.

Generalization of the theorem of de Montessus de Bollore

We prove a generalization of the theorem of de Montessus de Bollore for a large class of functional series investigated in [14]. In particular multivalued approximants for power series, as well as for series of Faber and Gegenbauer polynomials are considered. Numerical results are presented.

Minimization of Errors of the Polynomial-Trigonometric Interpolation with Shifted Nodes.

The polynomial-trigonometric interpolation based on the Krylov approach for a smooth function given on $[-1, 1]$ is defined on the union of $m$ shifted each other uniform grids with the equal number of points. The asymptotic errors of the interpolation in both uniform and $L_2$ metrics are investigated. It turned out that the corresponding errors can be minimized due to an optimal choice of the shift parameters. The study of asymptotic errors is based on the concept of the ''limit function" proposed by Vallee-Poussin.

In particular cases of unions of two and three uniform grids the limit functions are found explicitly and the optimal shift parameters are calculated using MATHEMATICA 4.1 computer system.

On the denoising problem

In various applications the problem on separation the original signal and the noise arises. In this paper we consider two cases, which naturally arise in applied problems. In the first case, the original signal permits linear prediction by its past behavior. In the second case the original signal is the values of some analytic function at a points from unit disk. In the both cases the noise is assumed to be a stationary process with zero mean value.

Let us note that the first case arises in Physical phenomena consideration. The second case arises in Identification problems for linear systems.

In this article we pose the problem of existence and uniqueness of convex body for which the projection curvature radius function coincides with given function. We and a necessary and sufficient condition that ensures a positive answer to both questions and suggest an algorithm of construction of the body. Also we find a representation of the support function of a convex body by projection curvature radii.

Weighted Classes of Regular Functions Area Integrable Over the Unit Disc

This preprint contains some generalizations of the main theorems of M.M.Djrbashian of 1945--1948 which laid ground for the theory of $A^p_\alpha$ $($or initially $H^p(\alpha))$ spaces and his factorization theory of classes $N\{\omega\}$ exhausting all functions meromorphic in the unit disc. Also some later results on $A^p_\alpha$ spaces and Nevanlinna's weighted class are improved.

The preprint contains the main analytic apparatus for generalizing almost all known results on $A^p_\alpha$ spaces within a new theory, where instead $(1-r^2)^\alpha dr$ $(-1<\alpha<+\infty$, $0<r<1)$ some weights of the form $d\omega(r^2)$ are used. The obtained results make evident that the theory of $A^p_\omega$ spaces and the factorization theory of M.M.Djrbashian are inseparable parts of a general theory of classes of regular functions associated with M.M.Djrbashian general integrodifferentiation. The author hopes that the publication of this preprint can lead to clarification of some priority misunderstandings in the field.